Abstract The kinematics of the Milky Way disc as a function of age are well measured at the solar radius, but have not been studied over a wider range of Galactocentric radii. Here, we measure the kinematics of mono-age, mono-[Fe/H] populations in the low and high [α/Fe] discs between 4 ≲ R ≲ 13 kpc and |z| ≲ 2 kpc using 65 719 stars in common between APOGEE DR14 and Gaia DR2 for which we estimate ages using a Bayesian neural network model trained on asteroseismic ages. We determine the vertical and radial velocity dispersions, finding that the low and high [α/Fe] discs display markedly different age–velocity dispersion relations (AVRs) and shapes σz/σR. The high [α/Fe] disc has roughly flat AVRs and constant σz/σR = 0.64 ± 0.04, whereas the low [α/Fe] disc has large variations in this ratio that positively correlate with the mean orbital radius of the population at fixed age. The high [α/Fe] disc component’s flat AVRs and constant σz/σR clearly indicate an entirely different heating history. Outer disc populations also have flatter radial AVRs than those in the inner disc, likely due to the waning effect of spiral arms. Our detailed measurements of AVRs and σz/σR across the disc indicate that low [α/Fe], inner disc ($R \lesssim 10\, \mathrm{kpc}$) stellar populations are likely dynamically heated by both giant molecular clouds and spiral arms, while the observed trends for outer disc populations require a significant contribution from another heating mechanism such as satellite perturbations. We also find that outer disc populations have slightly positive mean vertical and radial velocities likely because they are part of the warped disc.